Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,9,12)(7,8,11,10)(13,14,17,19)(15,22,24,20)(16,25,18,21)(23,27,26,28);(5,7,9,11)(6,10,12,8)(14,19)(15,23)(16,18)(20,27)(22,28)(24,26);(2,4)(5,8,9,10)(6,11,12,7)(13,15,17,24)(14,20,19,22)(16,23,18,26)(21,27,25,28);(2,4)(6,12)(7,11)(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,27,22,28)Polycyclic presentation:
2^8|a^2=efh,b^2=eh,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6634628, #6634631, #6634632, … |
| maximal normal subgroup of | #6634628, #6634631, #6634632, … |
| maximal quotient of | #345338, #345339, #345340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24194);